Search

Dual Bernstein polynomials of one or two variables have proved to be very useful in obtaining Bézier form of the L -solution of the problem of best polynomial approximation of Bézier curve or surface. In this connection, the Bézier coefficients of dual Bernstein polynomials are to be evaluated at a reasonable cost. In this paper, a set of recurrence relations satisfied by the Bézier coefficients of dual bivariate Bernstein polynomials is derived and an efficient algorithm for evaluation of these coefficients is proposed. Applications of this result to some approximation problems of Computer Aided Geometric Design (CAGD) are discussed. [ABSTRACT FROM AUTHOR]

We propose a novel approach to the problem of polynomial approximation of rational Bézier triangular patches with prescribed boundary control points. The method is very efficient thanks to using recursive properties of the bivariate dual Bernstein polynomials and applying a smart algorithm for evaluating a collection of two-dimensional integrals. Some illustrative examples are given. [ABSTRACT FROM AUTHOR]

This paper deals with the problem of multi-degree reduction of a composite Bézier curve with the parametric continuity constraints at the endpoints of the segments. We present a novel method which is based on the idea of using constrained dual Bernstein polynomials to compute the control points of the reduced composite curve. In contrast to other methods, ours minimizes the L 2 -error for the whole composite curve instead of minimizing the L 2 -errors for each segment separately. As a result, an additional optimization is possible. Examples show that the new method gives much better results than multiple application of the degree reduction of a single Bézier curve. [ABSTRACT FROM AUTHOR]

We present a new approach to the problem of G-constrained ( k, l ≤ 3) multi-degree reduction of Bézier curves with respect to the least squares norm. First, to minimize the least squares error, we consider two methods of determining the values of geometric continuity parameters. One of them is based on quadratic and nonlinear programming, while the other uses some simplifying assumptions and solves a system of linear equations. Next, for prescribed values of these parameters, we obtain control points of the multi-degree reduced curve, using the properties of constrained dual Bernstein basis polynomials. Assuming that the input and output curves are of degree n and m, respectively, we determine these points with the complexity O( m n), which is significantly less than the cost of other known methods. Finally, we give several examples to demonstrate the effectiveness of our algorithms. [ABSTRACT FROM AUTHOR]

Abstract: We give structure relations for the orthogonal polynomials in two variables, defined by Lewanowicz and Woźny [S. Lewanowicz, P. Woźny, J. Comput. Appl. Math. 233 (2010) 1554–1561] where , and are univariate big q-Jacobi polynomials. We discuss in full detail the case of the polynomials , which are closely related to Dunkl’s bivariate (little) q-Jacobi polynomials [C.F. Dunkl, SIAM J. Algebra. Discr. Methods 1 (1980) 137–151]. [Copyright &y& Elsevier]

We present an efficient method to solve the problem of the constrained least squares approximation of the rational Bézier curve by the polynomial Bézier curve. The presented algorithm uses the dual constrained Bernstein basis polynomials, and exploits their recursive properties. Examples are given, showing the effectiveness of the algorithm. [ABSTRACT FROM AUTHOR]

Abstract: Explicit formulae for the Bézier coefficients of the constrained dual Bernstein basis polynomials are derived in terms of the Hahn orthogonal polynomials. Using difference properties of the latter polynomials, efficient recursive scheme is obtained to compute these coefficients. Applications of this result to some problems of CAGD is discussed. [Copyright &y& Elsevier]

Abstract: We propose an efficient approach to the problem of multi-degree reduction of rectangular Bézier patches, with prescribed boundary control points. We observe that the solution can be given in terms of constrained bivariate dual Bernstein polynomials. The complexity of the method is with m ≔min(m 1, m 2), where (n 1, n 2) and (m 1, m 2) is the degree of the input and output Bézier surface, respectively. If the approximation—with appropriate boundary constraints—is performed for each patch of several smoothly joined rectangular Bézier surfaces, the result is a composite surface of global C r continuity with a prescribed r ⩾0. In the detailed discussion, we restrict ourselves to r ∈{0,1}, which is the most important case in practical application. Some illustrative examples are given. [ABSTRACT FROM AUTHOR]

Abstract: We propose a novel approach to the problem of multi-degree reduction of Bézier triangular patches with prescribed boundary control points. We observe that the solution can be given in terms of bivariate dual discrete Bernstein polynomials. The algorithm is very efficient thanks to using the recursive properties of these polynomials. The complexity of the method is , and being the degrees of the input and output Bézier surfaces, respectively. If the approximation—with appropriate boundary constraints—is performed for each patch of several smoothly joined triangular Bézier surfaces, the result is a composite surface of global continuity with a prescribed order . Some illustrative examples are given. [ABSTRACT FROM AUTHOR]

Abstract: A four-parameter family of orthogonal polynomials in two discrete variables is defined for a weight function of basic hypergeometric type. The polynomials, which are expressed in terms of univariate big q-Jacobi polynomials, form an extension of Dunkl’s bivariate (little) q-Jacobi polynomials [C.F. Dunkl, Orthogonal polynomials in two variables of -Hahn and -Jacobi type, SIAM J. Algebr. Discrete Methods 1 (1980) 137–151]. We prove orthogonality property of the new polynomials, and show that they satisfy a three-term relation in a vector-matrix notation, as well as a second-order partial q-difference equation. [Copyright &y& Elsevier]

Abstract: We present a novel approach to the problem of multi-degree reduction of Bézier curves with constraints, using the dual constrained Bernstein basis polynomials, associated with the Jacobi scalar product. We give properties of these polynomials, including the explicit orthogonal representations, and the degree elevation formula. We show that the coefficients of the latter formula can be expressed in terms of dual discrete Bernstein polynomials. This result plays a crucial role in the presented algorithm for multi-degree reduction of Bézier curves with constraints. If the input and output curves are of degree n and m, respectively, the complexity of the method is , which seems to be significantly less than complexity of most known algorithms. Examples are given, showing the effectiveness of the algorithm. [Copyright &y& Elsevier]

Abstract: Connection coefficients between the two-variable Bernstein and Jacobi polynomial families on the triangle are given explicitly as evaluations of two-variable Hahn polynomials. Dual two-variable Bernstein polynomials are introduced. Explicit formula in terms of two-variable Jacobi polynomials and a recurrence relation are given. [Copyright &y& Elsevier]

Abstract: The generalized Bernstein basis in the space of polynomials of degree at most n, being an extension of the q-Bernstein basis introduced by Philips [Bernstein polynomials based on the q-integers, Ann. Numer. Math. 4 (1997) 511–518], is given by the formula [S. Lewanowicz, P. Woźny, Generalized Bernstein polynomials, BIT Numer. Math. 44 (2004) 63–78], We give explicitly the dual basis functions for the polynomials , in terms of big q-Jacobi polynomials , a and b being parameters; the connection coefficients are evaluations of the q-Hahn polynomials. An inverse formula—relating big q-Jacobi, dual generalized Bernstein, and dual q-Hahn polynomials—is also given. Further, an alternative formula is given, representing the dual polynomial as a linear combination of big q-Jacobi polynomials with shifted parameters and argument. Finally, we give a recurrence relation satisfied by , as well as an identity which may be seen as an analogue of the extended Marsden''s identity [R.N. Goldman, Dual polynomial bases, J. Approx. Theory 79 (1994) 311–346]. [Copyright &y& Elsevier]

This paper deals with the merging problem of segments of a composite Bézier curve, with the endpoints continuity constraints. We present a novel method which is based on the idea of using constrained dual Bernstein polynomial basis (Woźny and Lewanowicz, 2009) [12]to compute the control points of the merged curve. Thanks to using fast schemes of evaluation of certain connections involving Bernstein and dual Bernstein polynomials, the complexity of our algorithm is significantly less than complexity of other merging methods. [ABSTRACT FROM AUTHOR]